SOLVED: gPodder does this. I really should have known this.

I'm trying to be a little more deliberate about my media consumption, and less reliant on my phone, so I recently picked up a dedicated audio player. I have the music side sorted, but the other thing I listen to a lot is podcasts. So, what I'm looking for is a desktop podcast client that can download new episodes and track listens, but also transfer podcasts to the player. Somewhat like how desktop music players let you manage a device from within the app. From what I've seen, there's a lot of nice clients out there that do the first two, but I can't find anything that does the third one. Or alternatively, I don't have any particular loyalty to desktop music players, so one of those with device management that can also do podcasts would work too.

The player shows up as USB mass storage, so I'm not opposed to doing the syncing via a file manager or script, I'd just prefer not to have to.

EDIT: Linux